Step: 3 Underline the unit digit in Column II and add the number formed by the tens and other digits if any, with a2 in column I.

Column I

Column II

Column III

a2

2 x a x b

b2

81 + 11

108 + 3

36

92

111

Step: 4 underline the number in Column I

Column I

Column II

Column III

a2

2 x a x b

b2

81 + 11

108 + 3

36

92

111

Step: 5 write the underlined digits at the bottom of each column to obtain the square of the given number.

In this case, we have:

962 = 9216

Using multiplication:

96 x 96 = 2916

This matches with the result obtained using the column method.

Question: 2

Find the squares of the following numbers using diagonal method:

(i) 98

(ii) 273

(iii) 348

(iv) 295

(v) 171

Solution:

(i) 98

∴ 982 = 9604

(ii) 273

∴ 2732 = 74529

(iii) 348

∴ 3482 = 121104

(iv) 295

∴ 2952 = 87025

(v) 171

∴ 1712 = 29241

Question: 3

Find the squares of the following numbers:

(i) 127

(ii) 503

(iii) 451

(iv) 862

(v) 265

Solution:

We will use visual method as it is the efficient method to solve this problem.

(i) We have:

127 = 120 + 7

Hence, let us draw a square having side 127 units. Let us split it into 120 units and 7 units.

Hence, the square of 127 is 16129.

(ii) We have:

503 = 500 + 3

Hence, let us draw a square having side 503 units. Let us split it into 500 units and 3 units.

Hence, the square of 503 is 253009.

(iii) We have:

451 = 450 + 1

Hence, let us draw a square of having side 451 units. Let us split it into 450 units and 1 units.

Hence, the square of 451 is 203401.

(iv) We have:

862 = 860 + 2

Hence, let us draw a square having side 862 units. Let us split it into 860 units and 2 units.

Hence, the square of 862 is 743044.

(v) We have:

265 = 260 +5

Hence, let us draw a square having 265 units. Let us split it into 260 units and 5 units.

Hence, the square of 265 units is 70225.

Question: 4

Find the squares of the following numbers:

(i) 425

(ii) 575

(iii) 405

(iv) 205

(v) 95

(vi) 745

(vii) 512

(viii) 995

Notice that all numbers except the one in question (vii) has 5 as their respective unit digits. We know that the square of a number with the form n5 is a number ending with 25 and has the number n(n + 1) before 25.

Solution:

Sol.

(i) 425

Here, n = 42

∴ n(n + 1) = (42)(43) = 1806

∴ 4252 = 180625

(ii) 575

Here, n = 57

∴n(n + 1) = (57)(58) = 3306

∴ 5752 = 330625

(iii) 405

Here n = 40

∴n(n + 1) = (40)(41) = 1640

∴ 4052 = 164025

(iv) 205

Here n = 20

∴n(n + 1) = (20)(21) = 420

∴ 2052 = 42025

(v) 95

Here n = 9

∴ n(n + 1) = (9)(10) = 90

∴ 952 = 9025

(vi) 745

Here n = 74

∴ n(n + 1) = (74)(75) = 5550

∴ 7452 = 555025

(vii) 512

We know: The square of a three-digit number of the form 5ab = (250 + ab) 1000 + (ab)2

∴ 5122 = (250+12)1000 + (12)2 = 262000 + 144 = 262144

(viii) 995

Here, n = 99

∴ n(n + 1) = (99)(100) = 9900

∴ 9952 = 990025

Question: 5

Find the squares of the following numbers using the identity (a +b)2 = a2 + 2ab + b2: